GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Dirac-Operator</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Dirac-Operator</span></h1>

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Der <b>Dirac-Operator</b> ist ein <a href="Differentialoperator" title="Differentialoperator">Differentialoperator</a>, der eine <a href="Quadratwurzel" title="Quadratwurzel">Quadratwurzel</a> aus dem <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a> ist. Der ursprüngliche Fall, mit dem sich <a href="Paul_Dirac" title="Paul Dirac">Paul Dirac</a> beschäftigte, war die formale <a href="Faktorisierung" title="Faktorisierung">Faktorisierung</a> eines Operators für den <a href="Minkowski-Raum" title="Minkowski-Raum">Minkowski-Raum</a>, der die Quantentheorie mit der <a href="Spezielle_Relativit%C3%A4tstheorie" title="Spezielle Relativitätstheorie">speziellen Relativitätstheorie</a> verträglich macht.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D\in \operatorname {Diff} ^{1}(V,V)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2ee5a2ca105d3cf8926411b18e62425fdb78579.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.083ex; height:3.176ex;" alt="{\displaystyle D\in \operatorname {Diff} ^{1}(V,V)}" loading="lazy"></span> ein <a href="Geometrischer_Differentialoperator" class="mw-redirect" title="Geometrischer Differentialoperator">geometrischer Differentialoperator</a> erster Ordnung, der auf ein <a href="Vektorb%C3%BCndel" title="Vektorbündel">Vektorbündel</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\to M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/861711b5ff0179ddda858cb4a7e9dc7e75354a5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.844ex; height:2.176ex;" alt="{\displaystyle V\to M}" loading="lazy"></span> über einer <a href="Riemannsche_Mannigfaltigkeit" title="Riemannsche Mannigfaltigkeit">riemannschen Mannigfaltigkeit</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> wirkt. Wenn dann

</p>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D^{2}=\Delta \,,}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8f707bdebd9674765a5e4373b1ed0866b74f4ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.047ex; height:3.009ex;" alt="{\displaystyle D^{2}=\Delta \,,}" loading="lazy"></span></dd></dl>

<p>gilt, wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"></span> ein <a href="Verallgemeinerter_Laplace-Operator" title="Verallgemeinerter Laplace-Operator">verallgemeinerter Laplace-Operator</a> auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> ist, so heißt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> Dirac-Operator.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Geschichte">Geschichte</h2></div>

<p>Ursprünglich hatte <a href="Paul_Dirac" title="Paul Dirac">Paul Dirac</a> die Wurzel aus dem <a href="D%E2%80%99Alembertoperator" class="mw-redirect" title="D’Alembertoperator">D’Alembertoperator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \square }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \square }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/455831d58fa08f311b934d324adcff89a868b4e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \square }" loading="lazy"></span> betrachtet und damit die relativistische Quantenfeldtheorie eines <a href="Elektron" title="Elektron">Elektrons</a> begründen wollen.

</p><p>Dirac betrachtete für n=3 den Differentialoperator

</p>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}\gamma _{i}{\frac {\partial }{\partial x_{i}}}\,,}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05502bea379f7b281686bb4399eb25f95badef49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.064ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}\gamma _{i}{\frac {\partial }{\partial x_{i}}}\,,}" loading="lazy"></span></dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma _{i}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/907e4a28946d45e5cbf3cf3c6c48de68296039c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.004ex; height:2.176ex;" alt="{\displaystyle \gamma _{i}}" loading="lazy"></span> die <a href="Dirac-Matrizen" title="Dirac-Matrizen">Dirac-Matrizen</a> sind. Dieser ist jedoch nach heutigem Verständnis kein Dirac-Operator mehr.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>In den 1960ern griffen <a href="Michael_Francis_Atiyah" title="Michael Francis Atiyah">Michael Francis Atiyah</a> und <a href="Isadore_M._Singer" title="Isadore M. Singer">Isadore M. Singer</a> diesen von Dirac definierten Differentialoperator auf und entwickelten daraus den hier im Artikel hauptsächlich beschriebenen (verallgemeinerten) Dirac-Operator. Der Name Dirac-Operator wurde von Atiyah und Singer geprägt. Der Operator beeinflusste die Mathematik und die <a href="Mathematische_Physik" title="Mathematische Physik">mathematische Physik</a> des 20. Jahrhunderts stark.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Der_Dirac-Operator_eines_Dirac-Bündels"><span id="Der_Dirac-Operator_eines_Dirac-B.C3.BCndels"></span>Der Dirac-Operator eines Dirac-Bündels</h2></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (M,g)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e27d2e539fd0c3a9a7efab6257abd17de7fc57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="{\displaystyle (M,g)}" loading="lazy"></span> eine riemannsche Mannigfaltigkeit und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {E}},h,\nabla ^{\mathcal {E}})}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {E}},h,\nabla ^{\mathcal {E}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d0fae0556a3ff06a1f6fa59fc07518ab7d8d661.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.622ex; height:3.176ex;" alt="{\displaystyle ({\mathcal {E}},h,\nabla ^{\mathcal {E}})}" loading="lazy"></span> ein Dirac-Bündel, bestehend aus einem <a href="Clifford-Modul-B%C3%BCndel" title="Clifford-Modul-Bündel">Clifford-Modul-Bündel</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}\to M}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}\to M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3044b8e8436c1a674e87f8414c049f43042e0d42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.367ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}\to M}" loading="lazy"></span> einer hermiteschen Metrik <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c298ed828ff778065aeb5f0f305097f55bb9ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}" loading="lazy"></span> und einem Clifford-Zusammenhang <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{\mathcal {E}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla ^{\mathcal {E}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1282e3d47069ce038fb81bbfa454c6dd3a76540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.676ex;" alt="{\displaystyle \nabla ^{\mathcal {E}}}" loading="lazy"></span> auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c298ed828ff778065aeb5f0f305097f55bb9ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}" loading="lazy"></span>. Dann ist der Operator

</p>

</mrow>

<mover>

<msup>

</mrow>

</msup>

</mpadded>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<mover>

</mpadded>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D\colon \Gamma (M,{\mathcal {E}}){\xrightarrow {\nabla ^{\mathcal {E}}}}\Gamma (M,T^{*}M\otimes {\mathcal {E}}){\xrightarrow {c}}\Gamma (M,{\mathcal {E}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3698632fe763744f03496e10d81ef606a5f844a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-top: -0.402ex; width:41.096ex; height:4.676ex;" alt="{\displaystyle D\colon \Gamma (M,{\mathcal {E}})\xrightarrow {\nabla ^{\mathcal {E}}} \Gamma (M,T^{*}M\otimes {\mathcal {E}})\xrightarrow {c} \Gamma (M,{\mathcal {E}})}" loading="lazy"></span></dd></dl>

<p>der zum Dirac-Bündel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,h,\nabla ^{\mathcal {E}})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,h,\nabla ^{\mathcal {E}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4904a6d8ea26fe1741ce716c40bbead22708077d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.087ex; height:3.176ex;" alt="{\displaystyle (E,h,\nabla ^{\mathcal {E}})}" loading="lazy"></span> assoziierte Dirac-Operator. In lokalen Koordinaten hat er die Darstellung

</p>

</mrow>

</mrow>

</munderover>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=\sum _{i=1}^{n}c(\mathrm {d} x^{i})\nabla _{\partial _{i}}^{\mathcal {E}}\,.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc4e4e3ed4cd5d11610c7b0a1d6ad4a723072c88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.703ex; height:6.843ex;" alt="{\displaystyle D=\sum _{i=1}^{n}c(\mathrm {d} x^{i})\nabla _{\partial _{i}}^{\mathcal {E}}\,.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Elementares_Beispiel">Elementares Beispiel</h3></div>

<p>Der Operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\partial _{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -i\partial _{x}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e64e95e8c209a109cb9be9e09f5222c70d6707ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.018ex; height:2.509ex;" alt="{\displaystyle -i\partial _{x}}" loading="lazy"></span> ist ein Dirac-Operator über dem <a href="Tangentialb%C3%BCndel" title="Tangentialbündel">Tangentialbündel</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Spin-Dirac-Operator">Spin-Dirac-Operator</h3></div>

<p>Betrachtet werde der Konfigurationsraum eines Teilchens mit <a href="Spin" title="Spin">Spin</a> <sup>1</sup>/<sub>2</sub>, das auf die Ebene <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"></span> beschränkt ist, welche die Basis-Mannigfaltigkeit bildet. Der Zustand wird durch eine Wellenfunktion ψ<sub>G</sub> mit zwei komplexen Komponenten beschrieben, für die also jeweils <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}\to \mathbb {C} \,}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}\to \mathbb {C} \,}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d6993feca9248f470ea6f6a02f10a0c10092b58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.412ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}\to \mathbb {C} \,}" loading="lazy"></span> gelten soll, wobei Gesamtzustände, die sich nur um einen komplexen Faktor unterscheiden, identifiziert werden. Der Gesamtzustand ist also:

</p>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{G}={\begin{bmatrix}\chi _{\uparrow }(x,y)\\\eta _{\downarrow }(x,y)\end{bmatrix}}\,.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f39725f829272070f9f60e94afceb545611aaaff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.214ex; height:6.509ex;" alt="{\displaystyle \psi _{G}={\begin{bmatrix}\chi _{\uparrow }(x,y)\\\eta _{\downarrow }(x,y)\end{bmatrix}}\,.}" loading="lazy"></span></dd></dl>

<p>Dabei sind <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> die üblichen kartesischen Koordinaten auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"></span>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{\uparrow }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{\uparrow }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20c2c0cfea19858a19c1eaea51afc2f52c43d8cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.509ex; height:2.343ex;" alt="{\displaystyle \chi _{\uparrow }}" loading="lazy"></span> definiert die Wahrscheinlichkeitsamplitude für die aufwärts gerichteten Spin-Komponente (Spin-Up), und analog <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\downarrow }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta _{\downarrow }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25a261473d3deec083d5fb19222576d71f218424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.21ex; height:2.343ex;" alt="{\displaystyle \eta _{\downarrow }}" loading="lazy"></span> für die Spin-Down-Komponente. Der sogenannte Spin-Dirac-Operator kann dann geschrieben werden als

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=-i\sigma _{x}\partial _{x}-i\sigma _{y}\partial _{y},}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6f435803189f3ccdef788e55cd536acad4863ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.49ex; height:2.843ex;" alt="{\displaystyle D=-i\sigma _{x}\partial _{x}-i\sigma _{y}\partial _{y},}" loading="lazy"></span></dd></dl>

<p>wobei σ<sub><i>x</i></sub> und σ<sub><i>y</i></sub> die <a href="Pauli-Matrizen" title="Pauli-Matrizen">Pauli-Matrizen</a> sind. Man beachte, dass die <a href="Antikommutativit%C3%A4t" class="mw-redirect" title="Antikommutativität">antikommutativen</a> Beziehungen der Pauli-Matrizen einen Beweis der obigen Definition trivial machen. Diese Beziehungen definieren den Begriff der <a href="Clifford-Algebra#Beispiele" title="Clifford-Algebra">Clifford-Algebra#Beispiele</a> am Beispiel der <a href="Quaternion" title="Quaternion">Quaternionen-Algebra</a>. Lösungen der <a href="Dirac-Gleichung" title="Dirac-Gleichung">Dirac-Gleichung</a> für Spinor-Felder werden oft <i>harmonische Spinoren</i> genannt<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Hodge-De-Rham-Operator">Hodge-De-Rham-Operator</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (M,g)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e27d2e539fd0c3a9a7efab6257abd17de7fc57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="{\displaystyle (M,g)}" loading="lazy"></span> eine <a href="Orientierung_(Mathematik)" title="Orientierung (Mathematik)">orientierbare</a> <a href="Riemannsche_Mannigfaltigkeit" title="Riemannsche Mannigfaltigkeit">riemannsche Mannigfaltigkeit</a> und sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \colon {\mathcal {A}}(M)^{\bullet -1}\to {\mathcal {A}}^{\bullet }(M)}">

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \colon {\mathcal {A}}(M)^{\bullet -1}\to {\mathcal {A}}^{\bullet }(M)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b413eef72b6ccbcaaa6921b97fe79a4767692b22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.474ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} \colon {\mathcal {A}}(M)^{\bullet -1}\to {\mathcal {A}}^{\bullet }(M)}" loading="lazy"></span> die <a href="%C3%84u%C3%9Fere_Ableitung" title="Äußere Ableitung">äußere Ableitung</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} ^{t}\colon {\mathcal {A}}^{\bullet }(M)\to {\mathcal {A}}^{\bullet -1}(M)}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} ^{t}\colon {\mathcal {A}}^{\bullet }(M)\to {\mathcal {A}}^{\bullet -1}(M)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58847e542455b5381bf2ca16044931ba7fc3ead3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.315ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} ^{t}\colon {\mathcal {A}}^{\bullet }(M)\to {\mathcal {A}}^{\bullet -1}(M)}" loading="lazy"></span> der zur äußeren Ableitung bezüglich der L²-Metrik <a href="Adjungierter_Operator" title="Adjungierter Operator">adjungierte Operator</a>. Dann ist

</p>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} +\mathrm {d} ^{t}\colon {\mathcal {A}}^{\bullet }(M)\to {\mathcal {A}}^{\bullet }(M)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/556deed483dc194a25f58acc79481039f5236e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.347ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} +\mathrm {d} ^{t}\colon {\mathcal {A}}^{\bullet }(M)\to {\mathcal {A}}^{\bullet }(M)}" loading="lazy"></span></dd></dl>

<p>ein Dirac-Operator.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Atiyah-Singer-Dirac-Operator">Atiyah-Singer-Dirac-Operator</h3></div>

<p>Es gibt auch einen Dirac-Operator in der Clifford-Analysis. Im n-dimensionalen euklidischen Raum, d. h. für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{n}\,,}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{n}\,,}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4025c114f7612c0ef919ae6fa945c166f547e05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.441ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{n}\,,}" loading="lazy"></span> ist das<br>    <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=\sum _{j=1}^{n}e_{j}{\frac {\partial }{\partial x_{j}}}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=\sum _{j=1}^{n}e_{j}{\frac {\partial }{\partial x_{j}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2077877e84738c465583443923b72db8c34a0f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.152ex; height:7.176ex;" alt="{\displaystyle D=\sum _{j=1}^{n}e_{j}{\frac {\partial }{\partial x_{j}}}}" loading="lazy"></span><br>wobei<br>    <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{j}:j=1,\ldots ,n\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{e_{j}:j=1,\ldots ,n\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/572ea6ff885fab74acbdd384047b676693082f22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.047ex; height:3.009ex;" alt="{\displaystyle \{e_{j}:j=1,\ldots ,n\}}" loading="lazy"></span><br>eine Orthonormal-Basis des euklidischen Raumes ist und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span> in eine <a href="Clifford-Algebra" title="Clifford-Algebra">Clifford-Algebra</a> eingebettet ist. Dies ist ein Spezialfall des Atiyah-Singer-Dirac-Operators, der auf den Schnitten eines <a href="Spinor#Verallgemeinerung_in_der_Mathematik" title="Spinor">Spinor-Bündels</a> wirkt.

</p><p>Für eine <a href="Spin-Mannigfaltigkeit" class="mw-redirect" title="Spin-Mannigfaltigkeit">Spin-Mannigfaltigkeit</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>, ist der Atiyah-Singer-Dirac-Operator lokal folgendermaßen definiert:<br>Für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1}(x),\ldots ,e_{j}(x)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{1}(x),\ldots ,e_{j}(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f900ee69d38f62e55251e2a8e48d971e58fb6b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.587ex; height:3.009ex;" alt="{\displaystyle e_{1}(x),\ldots ,e_{j}(x)}" loading="lazy"></span> eine lokale <a href="Orthonormalbasis" title="Orthonormalbasis">Orthonormalbasis</a> für den Tangentenraum von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> ist der Atiyah-Singer-Dirac-Operator<br>    <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=1}^{n}e_{j}(x){\tilde {\Gamma }}_{e_{j}(x)}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{j=1}^{n}e_{j}(x){\tilde {\Gamma }}_{e_{j}(x)}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b914fa250fe990e1b4542e3af60052229c4cf1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.259ex; height:7.176ex;" alt="{\displaystyle \sum _{j=1}^{n}e_{j}(x){\tilde {\Gamma }}_{e_{j}(x)}}" loading="lazy"></span>,<br>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\Gamma }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {\Gamma }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c4d49518192c4e8fb2514ff0703f2eb567d189a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.676ex;" alt="{\displaystyle {\tilde {\Gamma }}}" loading="lazy"></span> ein <a href="Paralleltransport" title="Paralleltransport">Paralleltransport</a> des <a href="Levi-Civita-Zusammenhang" title="Levi-Civita-Zusammenhang">Levi-Civita-Zusammenhangs</a> auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> für das Spinor-Bündel über <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> ist.

</p>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

<p>Das <a href="Hauptsymbol" class="mw-redirect" title="Hauptsymbol">Hauptsymbol</a> eines verallgemeinerten Laplace-Operators ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \mapsto \|\xi \|^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \mapsto \|\xi \|^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bce946212d85dd8c7c75d7dd87726e824eec9c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.053ex; height:3.176ex;" alt="{\displaystyle \xi \mapsto \|\xi \|^{2}}" loading="lazy"></span>. Entsprechend ist das Hauptsymbol eines Dirac-Operators <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \mapsto \|\xi \|}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \mapsto \|\xi \|}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b5404df9a25bcd9776b1f8ca4a5f13a568e8d26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.999ex; height:2.843ex;" alt="{\displaystyle \xi \mapsto \|\xi \|}" loading="lazy"></span> und somit sind beide Klassen von Differentialoperatoren <a href="Elliptischer_Differentialoperator" class="mw-redirect" title="Elliptischer Differentialoperator">elliptisch</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerungen">Verallgemeinerungen</h2></div>

<p>Der Operator

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D\colon C^{\infty }(\mathbb {R} ^{k}\otimes \mathbb {R} ^{n},S)\to C^{\infty }(\mathbb {R} ^{k}\otimes \mathbb {R} ^{n},\mathbb {C} ^{k}\otimes S)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6c9047790222eed3079cd45153f90432c72ab91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.218ex; height:3.176ex;" alt="{\displaystyle D\colon C^{\infty }(\mathbb {R} ^{k}\otimes \mathbb {R} ^{n},S)\to C^{\infty }(\mathbb {R} ^{k}\otimes \mathbb {R} ^{n},\mathbb {C} ^{k}\otimes S)}" loading="lazy"></span>, der

auf die nachfolgend definierten spinorwertige Funktionen wirkt,

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

<msub>

</mrow>

</msub>

</munder>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

<msub>

</mrow>

</msub>

</munder>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

<msub>

</mrow>

</msub>

</munder>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{1},\ldots ,x_{k})\mapsto {\begin{pmatrix}\partial _{\underline {x_{1}}}f\\\partial _{\underline {x_{2}}}f\\\ldots \\\partial _{\underline {x_{k}}}f\\\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1049d6b0c92326477ead9410e33c7afc9ac59e3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.337ex; margin-bottom: -0.668ex; width:26.048ex; height:14.676ex;" alt="{\displaystyle f(x_{1},\ldots ,x_{k})\mapsto {\begin{pmatrix}\partial _{\underline {x_{1}}}f\\\partial _{\underline {x_{2}}}f\\\ldots \\\partial _{\underline {x_{k}}}f\\\end{pmatrix}}}" loading="lazy"></span></dd></dl>

<p>wird in der Clifford-Analysis oft als Dirac-Operator in <i>k</i> Clifford-Variablen genannt. In dieser Notation ist <i>S</i> der Raum von <a href="Spinor" title="Spinor">Spinoren</a>,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=(x_{i1},x_{i2},\ldots ,x_{in})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf026ebe3e302d08672381f782b0fc7c86e8480b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.268ex; height:2.843ex;" alt="{\displaystyle x_{i}=(x_{i1},x_{i2},\ldots ,x_{in})}" loading="lazy"></span> sind <i>n</i>-dimensionale Variablen und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \partial _{\underline {x_{i}}}=\sum _{j}e_{j}\cdot \partial _{x_{ij}}}">

<msub>

<msub>

</mrow>

</msub>

</munder>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \partial _{\underline {x_{i}}}=\sum _{j}e_{j}\cdot \partial _{x_{ij}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8342512369b6d3eda0fc891be039de654f12492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.041ex; margin-bottom: -0.63ex; width:17.137ex; height:3.676ex;" alt="{\displaystyle \textstyle \partial _{\underline {x_{i}}}=\sum _{j}e_{j}\cdot \partial _{x_{ij}}}" loading="lazy"></span>

ist der Dirac-Operator in der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>-ten Variablen. Dies ist eine gebräuchliche Verallgemeinerung des Dirac-Operators (<i>k=1</i>) und der

<a href="Dolbeault-Kohomologie" title="Dolbeault-Kohomologie">Dolbeault-Kohomologie</a> (<i>n=2</i>, <i>k</i> beliebig). Er ist ein Differentialoperator, der invariant zu der Operation der Gruppe

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {SL} (k)\times \operatorname {Spin} (n)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aba2b883201abfbab242dfd814b9779b20c0042.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.335ex; height:2.843ex;" alt="{\displaystyle \operatorname {SL} (k)\times \operatorname {Spin} (n)}" loading="lazy"></span> ist. Die <a href="Injektive_Aufl%C3%B6sung" title="Injektive Auflösung">Injektive Auflösung</a> von <i>D</i> ist nur für einige Spezialfälle bekannt.

</p>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Atiyah-Singer-Indexsatz" title="Atiyah-Singer-Indexsatz">Atiyah-Singer-Indexsatz</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Thomas_Friedrich_(Mathematiker)" title="Thomas Friedrich (Mathematiker)">Thomas Friedrich</a>: <i>Dirac Operators in Riemannian Geometry</i> (Dirac-Operatoren in der Riemannschen Geometrie, 1997). <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, Providence, R.I. 2000, ISBN 978-0-8218-2055-1.</li>

<li>Fabrizio Colombo, Irene Sabadini: <i>Analysis of Dirac Systems and Computational Algebra</i> (Progress in mathematical physics; Bd. 39). Birkhäuser, Boston, Mass. 2004, ISBN 978-0-8176-4255-6.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Liviu I. Nicolaescu: <i>Lectures on the geometry of manifolds.</i> 2nd edition. World Scientific Pub Co., Singapore u. a. 2007, ISBN 978-981-270-853-3, S. 498</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Herbert Schröder: <cite style="font-style:italic">Funktionalanalysis</cite>. 2. korr. Auflage. Harri Deutsch, 2000, ISBN 3-8171-1623-3, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>364</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dirac-Operator&rft.au=Herbert+Schr%C3%B6der&rft.btitle=Funktionalanalysis&rft.date=2000&rft.edition=2.+korr.&rft.genre=book&rft.isbn=3817116233&rft.pages=364&rft.pub=Harri+Deutsch" style="display:none"> </span></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Yanlin Yu: <cite style="font-style:italic">The index theorem & the heat equation method</cite>. 1. Auflage. World Scientify, Singapur 2001, ISBN 981-02-4610-2, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>195</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dirac-Operator&rft.au=Yanlin+Yu&rft.btitle=The+index+theorem+%26+the+heat+equation+method&rft.date=2001&rft.edition=1.&rft.genre=book&rft.isbn=9810246102&rft.pages=195&rft.place=Singapur&rft.pub=World+Scientify" style="display:none"> </span></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">D. V. Alekseevskii (originator): <i><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Spinor_structure">Spinor structure</a>.</i> Encyclopedia of Mathematics</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Liviu I. Nicolaescu: <i>Lectures on the geometry of manifolds.</i> 2nd edition. World Scientific Pub Co., Singapore u. a. 2007, ISBN 978-981-270-853-3, S. 499</span>

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-05-12" href="https://de.wikipedia.org/wiki/?title=Dirac-Operator&oldid=255941602">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>